A175182 Pisano period of the 3-Fibonacci numbers A006190.

1, 3, 2, 6, 12, 6, 16, 12, 6, 12, 8, 6, 52, 48, 12, 24, 16, 6, 40, 12, 16, 24, 22, 12, 60, 156, 18, 48, 28, 12, 64, 48, 8, 48, 48, 6, 76, 120, 52, 12, 28, 48, 42, 24, 12, 66, 96, 24, 112, 60, 16, 156, 26, 18, 24, 48, 40, 84, 24, 12, 30, 192, 48, 96, 156, 24, 136, 48, 22, 48, 144

Offset: 1

Author

R. J. Mathar, Mar 01 2010

Keywords

• nonn
• easy

Comments

Period of the sequence defined by reading A006190 modulo n.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Sergio Falcon and Ángel Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41 (2009), 497-504.
• Eric Weisstein's World of Mathematics, Pisano period.
• Wikipedia, Pisano period.

Crossrefs

Cf. A001175, A006190, A175181, A175183, A175184, A175185.

Programs

Maple

F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc:
Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc:
k := 3 ; seq( Pper(k,m),m=1..80) ;

Mathematica

Table[s = t = Mod[{0, 1}, n]; cnt = 1; While[tmp = Mod[3*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}] (* Vincenzo Librandi, Dec 20 2012, T. D. Noe *)